In this reporting period publications appeared in the following areas: 1. Single-molecule fluorescence spectroscopy; 2. Single-molecule force spectroscopy; 3. Enzyme catalysis. These are discussed below in turn. Single-molecule Forster resonance energy transfer (FRET) between fluorescent donor and acceptor labels attached to a protein or nucleic acid is widely used to probe intramolecular distances and study the structure, dynamics and function of macromolecules. In these experiments, a molecule is illuminated by a laser, and the donor fluorophore is excited. The donor can emit a photon or transfer the excitation to an acceptor which then can emit a photon of a different color. The rate of transfer depends on the interdye distance and this is why there is information about conformational dynamics. The output of these experiments is a sequence of photons with recorded colors and arrival times. When a single molecule is excited by a pulsed laser, it is also possible to detect the time interval between the laser pulse and the photon. This so-called delay time is related to the fluorescence lifetime of the donor fluorophore. The distances between fluorescence labels attached to a molecule fluctuate due to conformational dynamics on a wide range of time scales. Extracting information about the dynamics is particularly challenging when the fluctuations are as fast as the time between photons. Previously, we developed methods of analyses of single-molecule photon sequences with recorded colors, arrival times, and delay times. The utility of these methods has been demonstrated on several fast-folding proteins, in collaboration with Dr. H.S. Chung from LCP. During this reporting period, in collaboration with Dr. H.S. Chung, we applied these and more challenging methods of three-color FRET to the oligomerization of the tetramerization domain (TD) of the tumor suppressor protein p53 (reference 1). TD exists as a monomer at subnanomolar concentrations and forms a dimer and a tetramer at higher concentrations. Because the dissociation constants of the dimer and tetramer are very close, as we determine in this paper, it is not possible to characterize different oligomeric species by ensemble methods. However, by combining two- and three-color single-molecule FRET spectroscopy with 2D FRET efficiencylifetime analysis, it is possible to determine structural information for individual oligomers at equilibrium and to determine the dimerization kinetics. From these analyses, we show that the monomer is intrinsically disordered and that the dimer conformation is very similar to that of the tetramer but the C terminus of the dimer is more flexible. The rate of energy transfer depends not only on the distance between the attached dyes but also on their relative orientation. Because one is interested in distances, this is a complication in using FRET to get conformational information. In order to develop a quantitative theory to address this problem, in reference 2, we derived a diffusion equation i for the time evolution of the orientational factor in the Forster energy transfer rate. The orientation-dependent diffusion coefficient is obtained in three different ways: (1) by requiring the orientational auto correlation function, calculated using the our one dimensional diffusion equation, to be single-exponential with the exact characteristic time; (2) by projecting the multidimensional diffusion equation for the transition dipoles using the local equilibrium approximation; and (3) by requiring exact and approximate trajectories to be as close as possible using a Bayesian approach. Within the framework of this theory, the distance dependence of the fluorescence resonance energy transfer (FRET) efficiency can be calculated for all values of the ratio of the rotational correlation time of the transition dipoles to the lifetime of the donor excited state. The theoretical predictions are in excellent agreement with the exact values obtained from Brownian dynamics simulations of the reorientation of the donor and acceptor transition dipoles. In single molecule force spectroscopy the response of an individual molecule to applied forces is probed using atomic force microscopes and laser tweezers. In reference 3, we solved the problem of how to interpret such experiments to yield information on the mechanical properties of biomolecules. Recall that ductile materials can absorb sudden spikes in mechanical force, whereas brittle ones fail catastrophically. Here we develop a theory to quantify the kinetic ductility of single molecules from force spectroscopy experiments, relating force-spike resistance to the differential responses of the intact protein and the unfolding transition state to an applied mechanical force. We introduce a class of unistable one-dimensional potential surfaces that encompass previous models as special cases and continuously cover the entire range from ductile to brittle. Compact analytic expressions for force-dependent rates and rupture-force distributions allow us to analyze force-clamp and force-ramp pulling experiments. We find that the force-transmitting protein domains of filamin and titin are kinetically ductile when pulled from their two termini, making them resistant to force spikes. For the mechanostable muscle protein titin, a highly ductile model reconciles data over 10 orders of magnitude in force loading rate from experiment and simulation. Enzymes are biological catalysts that play a fundamental role in all living systems by supporting the selectivity and the speed for almost all cellular processes. While the general principles of enzyme functioning are known, the specific details of how they work at the microscopic level are not always available. Simple Michaelis-Menten kinetics assumes that the enzyme-substrate complex has only one conformation that decays as a single exponential when the substrate dissociates. As a consequence, the enzymatic velocity decreases as the dissociation (off) rate constant of the complex increases. Recently, it was shown by a research group in Israel showed that it is possible for the enzymatic velocity to increase when the off rate becomes higher, if the enzyme-substrate complex has many conformations which dissociate with the same off rate constant. This was done using complicated formal mathematical arguments, without specifying the nature of the dynamics of the enzyme-substrate complex. In order to provide a physical basis for this unexpected result, in reference 4, we derive an analytical expression for the enzymatic velocity assuming that the enzyme-substrate complex has multiple states and its conformational dynamics is described by rate equations with arbitrary rate constants. By applying our formalism to a complex with two conformations, we show that the unexpected off rate dependence of the velocity can be readily understood: If one of the conformations is unproductive, the system can escape from this trap by dissociating, thereby giving the enzyme another. chance to form the productive enzyme-substrate complex.We also demonstrate that the nonmonotonic off rate dependence of the enzymatic velocity is possible not only when all off rate constants are identical, but even when they are different. We show that for typical experimentally determined rate constants, the nonmonotonic off rate dependence can occur for micromolar substrate concentrations. Finally, we discuss the relation of this work to the problem of optimizing the flux through singly occupied membrane channels and transporters.